ISSN 0862-5468 (Print), ISSN 1804-5847 (online) 

Ceramics-Silikáty 31, (4) 335 - 344 (1987)


RIEŠENIE STACIONÁRNEHO JEDNOROZMERNÉHO PRESTUPU TEPLA S UVAŽOVANÍM NELINEÁRNEJ TEPLOTNEJ ZÁVISLOSTI TEPELNEJ VODIVOSTI IZOLAČNÉHO MATERIÁLU V ZLOŽENEJ STENE

SOLUTION OF STATIONARY UNIDIMENSIONAL HEAT TRANSFER ASSUMING NON-LINEAR DEPENDENCE OF THERMAL CONDUCTIVITY OF INSULATING MATERIAL IN A SANDWICH WALL


 
Zábavník Viktor 1
 
1 Ústav výpočtovej techniky pri Vysokej škole technickej, ul. Boženy Němcovej, 040 00 Košice
1 Institute of Computer Technology at the Technical University, ul. Boženy Němcovej, 040 00 Košice

Stanovenie teplotného profilu v zloženej stene pre nelineárnu teplotnú závislosť tepelnej vodivosti na teplote vedie k riešeniu sústavy nelineárnych rovnic. Iteračný prístup značne zjednodušuje riešenie uvedenej úlohy. Definovaním rozdielnych spôsobov iterovania pri rovnakých podmienkach je možné voliť kompromis medzi presnostou a rýchlostou výpočtu. Z algoritmického hladiska ide o využitie dvoch viazaných iteračných cyklov. Napriek tejto skutočnosti ani pri vysokých požiadavkách na presnosť riešenia pri uvažovanom počte vrstiev 6-10 v zloženej stene neprekroči doba riešenia úlohy na počítači SM 3-4 minuty. Zmenou sposobu iterovania je možné tento čas znížit na tretinu až štvrtinu uvedenej hodnoty pri súčasnom zvýšení nepresnosti do 1 %.

Within the framework of technical accuracy, linear approximation of thermal conductivity in terms of temperature holds for most current refractory materials. Determination of the temperature profile in a sandwich wall by the iteration method leads to the solving of quadratic equations whose number is identical with the number of points having unknown temperatures. If the thermal conductivity has to be approximated in terms of temperature by a higher order plynomial, the problem is not resolved by means of a quadratic equation to determine the unknown temperature at the contact of the layers, being based instead on determining the roots of the polynomial the order of which is always higher by one compared to the polynomial for the approximation of thermal conductivity. Only one root among those seeked has physical significance. For this reason, the iteration method is advantageous for its determination. From the algorithmic point of view, it is in fact utilization of two bound interaction cycles, where the temperature at the contact of the layers is determined in the inner cycle on the basis of the quantities assumed. The outer iteration cycle is used to adjust the assumption to fit the boundary conditions. The procedure represents precise solution of the problem from the point of view of thermal technology. For a sandwich wall consisting of six to ten layers, the not time of solution on the SM 3 computer amounts to 2-4 minutes (excluding the input and output operations). An approximate method was suggested in order to further cut the time of the computation. It is based on the assumption that the thermal conductivity is constant per partes. Its actual value is computed by means of the non-linear temperature dependence of thermal conductivity. In this way, the time of computation is reduced approximately to one third or one quarter compared to the precise solution under identical iteration conditions. The resulting deviations from the exact solution do not exceed 1 %. The significance of the approximate solution increases particularly in the utilization of the solutfon in complex simulation systems.


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